• Introduction instanton molecules and topological susceptibility

• Random matrix model• Chiral condensate and Dirac spectrum• A modified model and Topological susceptibility • Summary

Topological susceptibility at finite Topological susceptibility at finite temperaturetemperature

in a random matrix modelin a random matrix model

Chiral 07, 14 Nov. @ RCNP

Munehisa Ohtani (Univ. Regensburg) with C. Lehner, T. Wettig (Univ.

Regensburg)

T. Hatsuda (Univ. of Tokyo)

IntroductionIntroduction

_ Banks-Casher rel: = (0)

where () = 1/V (n) = 1/ Im Tr( D+i)1

E.-M.Ilgenfritz & E.V.Shuryak PLB325(1994)263

Chiral symmetry breaking and instanton molecules

_

_

: chiral restoration

# of I-I : Formation of instanton molecules

?

?

Index Theorem: 1 tr FF = N+ N

~

32 2

0 mode of +() chirality associatedwith an isolated (anti-) instanton

quasi 0 modes begin to have a non-zero eigenvalue

(0) becomes sparse

Instanton molecules &Topological susceptibilityInstanton molecules &Topological susceptibility

topological charge density q(x)

q(x)2

isolated (anti-)instantonsat low T

d4x q(x)2 decreases as T

d4x q(x)2 1/V d4yd4x(q(x)2 q(y)2 )/2 1/V

d4yd4x q(x)q(y) = Q2/V

The formation of instanton molecules suggests

decreasing topological susceptibility as T

(anti-)instanton moleculeat high T

q(x)2

Random matrix model at T 0

Chiral restoration and Topological susceptibility

A.D.Jackson & J.J.M.Verbaarschot, PRD53(1996)

Chiral symmetry: {DE , 5} = 0 Hermiticity: DE

†= DE

Random matrix modelRandom matrix model

ZQCD = det(iDE + mf ) YM / ///f

0 T The lowest Matsubara freq.

quasi 0 mode basis, i.e.

topological charge: Q = N+ N

with iDRM = 0 iW iW† 0

W C N × N +

ZRM = eQ2/2N DW eN/22trW†W det(iDRM + mf )

Q f

|

Hubbard Stratonovitch transformationHubbard Stratonovitch transformation T.Wettig, A.Schäfer, H.A.Weidenmüller, PLB367(1996)

1) ZRM rewritten with fermions integrate out random matrix W Action with 4-fermi int.

3) introduce auxiliary random matrix S CNf × Nf

integrate out

dim. of matrix N N+ N( V) plays a role of “1/ h ”

The saddle point eqs. for S, Q/N become exact in the thermodynamic limit.

ZRM = eQ2/2N DS eN /22trS†S det S + m iT

(N|Q|)/2

det(S + m)|Q|

Q iT S† + m

|

Chiral condensateChiral condensate

_

= m lnZRM /VNf =1 N tr S0 + m iT

1

whereS0 : saddle pt. value

VNf

iT

S0†

+ m

0.5

1

1.5

2

0.5

1

1.5

2

00.250.5

0.75

1

0.5

1

1.5

2

_ _

/ 0

T / Tc

m

The 2nd order transitionin the chiral limit

(Q = 0 at the saddle pt.)

0

0.5

1

1.5

2

2

0

2

0

0.25

0.5

0.75

1

0

0.5

1

1.5

2

()

T / Tc

Eigenvalue distribution of Dirac operatorEigenvalue distribution of Dirac operator _

() = 1/V (n) = 1/ Im Tr( D+i)1 = 1/Re|m i _

= m lnZ /VNf = Tr( iD+m)1

(

(0) becomes

sparse as T

instanton molecule

?

T/Tc

as N

Q2

= 1

1

N 2

Suppression of topological susceptibility Suppression of topological susceptibility

ln Z(Q)/Z(0) = Q2 / NQ4 / N3|Q| Q2 / NQ3 / N2Expansion by Q / N :

× 1

1 0 (as N )

2 N sinh /2

in RMM for

m

Q / N

ln Z(Q)/Z(0)

Q / N m

Unphysical suppressionof at T in RMM

Leutwyler-Smilga model and Random Matrix Leutwyler-Smilga model and Random Matrix

Using singular value decomposition of S + m V1UV, ZRM is rewritten

with the part. func. ZL-S of chiral eff. theory for 0-momentum Goldstone modes

ZRM(Q) = NQ DZL-S(Q,) eN/2 2tr2det(2 + 2T2)N/2 det(2 + 2T2)|Q|/2

det|Q|

ZL-S(Q,) = DU eN 2trRe mUQ2/2N detUQ

H.Leutwyler, A.Smilga, PRD46(1992)

This factor suppresses We claim to

tune NQ so as to cancel the factor at the saddle point.

Modified Random Matrix modelModified Random Matrix model

ZmRM = DZL-S(Q,) eN/2 2tr2det(2 + 2T2)N/2

Q

We propose a modified model:

where _ in the conventional model is reproduced.

cancelled factor =1 at Q = 0 i.e. saddle pt. eq. does not change

(

at T = 0 in the conventional model is reproduced.

(

cancelled factor =1 also at T = 0 i.e. quantities at T = 0 do not change

at T > 0 is not suppressed in the thermodynamic limit.

T / Tc

m

topological susceptibility in the modified model topological susceptibility in the modified model

1

+ Nf

11

m(m+0)

where0 : saddle pt. value

m

m

· Decreasing as T · Comparable with lattice results

B.Alles, M.D’Elia, A.Di Giacomo, PLB483(2000)

Summary and outlookSummary and outlook Chiral restoration and topological susceptibility are studied in a random matrix model formation of instanton molecules connects them via Banks-Casher relation and the index theorem.

Conventional random matrix model : 2nd order chiral transition &

unphysical suppression of for T >0 in the thermodynamic limit.

We propose a modified model in which

& are same as in the original model, at T >0 is well-defined and decreases as T increases.

consistent with instanton molecule formation, lattice results

Outlook: To find out the random matrix before H-S transformation from which the modified model are derived, Extension to finite chemical potential, Nf dependence …

_